Sampling theory Assignment Help

Sampling theory

A continuous time random signal is shown. Based on this a number of important concepts are shown below. The signal is continuous time signal with continuous amplitude. This type of signal is known as an analog signal.

x(t)↑

4

8

3

7

2

6

1

5

0

4

→Time

-1

3

-2

2

-3

1

-4

0

nT→

0

1T

2T

3T

4T

5T

6T

7T

8T

→Time

n→

0

1

2

3

4

5

6

7

8

x(n)→

5.5

2.8

3.8

5.3

1.5

4.6

8.4

6.9

7.3

←Sampled signal. Discrete-time

signal - time is discrete, amplitude is continuous.

5

2

3

5

1

4

7

6

7

←Quantized. Quantization noise

(error). Digital signal - both time and amplitude are discrete.

101

010

011

101

001

100

111

110

111

Encoded to 3 bits/sample.

↑Note this particular point exhibits saturation (out of

range). Rounded down to 7, not 8.

If we were to represent sample value with infinite precision then we would require registers and memory words of arbitrarily large size. But, because of a finite word length we round off the sample values. This introduces the quantization noise.

The process of generating a discrete-time signal from the analog signal is shown in block diagram. In the digital signal processing course we are mostly dealing with discrete-time rather than digital signals and systems, the latter is a subset of the former.

The 3 boxes shown in the above diagram can be represented by an analog to digital converter (ADC). A complete digital signal processing (DSP) system comprises of an ADC, a DSP algorithm

(for example a difference equation) and a digital to analog converter (DAC) which is shown below.

As the name implies discrete-time signals are defined only at discrete instants of time. Discrete-time signals can arise by sampling analog signals such as a voice signal or a temperature signal in telemetry. Discrete-time signals may also arise naturally, e.g., the number of cars sold on a specific day in a year, or the closing DJIA figure for a specific day of the year.

AT&T's T1 Stream The voice signal is band limited to 3.3 kHz, sampled at 8000 Hz (8000 samples per second), quantized and encoded into 8 bits per sample. Twenty four such voice channels are combined to form the T1 stream or signal.